GCSE :: Surds

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Last modified: 13th September 2021 England and Wales (no 1194954)
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Types of numbers !

Real numbers are any

Real Numbers possible decimal or
whole number.

Rational Numbers Irrational Numbers

are all numbers which are real numbers which

can be expressed as are not rational.
some fraction involving
integers (whole
numbers), e.g. , , -7.
Types of numbers

Click each number to

see where it goes in the
Real numbers Venn Diagram.

Rational numbers

Integers 3 0.7
.
π 1.3

√𝟐 -1
3
4 √𝟗 𝒆
(Click the blue boxes above)
What is a surd?
Vote on whether you think the following might be ‘surds’ or not surds.

√2 Not
a surd 
Surd

√9 Not asurd 
Surd

√5 Not
a surd 
Surd

√√7
3
1
4
Not asurd

Not
a surd

Surd


Surd

Therefore, can you think of a suitable definition for a surd?

A surd is a root of a number that cannot ?

be simplified to a rational number.
Laws of Surds
The only two things you need to know this topic…

√ 𝑎 × √ 𝑏=√ 𝒂𝒃 ?

√𝑎 =
√𝑏 √ ?
𝒂
𝒃

Basic Examples:

√ 3 × √2=√ 𝟔? √ 1 √𝟏 𝟏
= ?=
9 √𝟗 𝟑
√ 2
√
4 𝑥 =√ 𝟒 𝒙 =𝟐 𝒙
𝟐
?
Simplifying Surds

√ 8=√ 𝟒? √𝟐=𝟐? √ 𝟐
Fro Tip: Find the largest
Could we somehow use to break the 8 square factor of the number,
up in a way that one of the surds will and put that first.
simplify?

√ √ √
27= 𝟗 𝟑=𝟑 ?
√𝟑
√ 32=√ 𝟏𝟔 √ 𝟐=𝟒
? √𝟐
√ 50=√ 𝟐𝟓 √ 𝟐=𝟓
? √𝟐
√ 12= √ 𝟒 √ 𝟑=𝟐
? √𝟑
Test Your Understanding So Far

a
√ 24= √𝟒 √𝟔=𝟐? √𝟔
b
√ 75=√ 𝟐𝟓 √ 𝟑=𝟓
? √𝟑
c
√ 20=√ 𝟒 √ 𝟓=𝟐
? √𝟓
d
√ 48= √𝟏𝟔 √𝟑=𝟒
? √𝟑

Practise this specific Key Skill:

https://www.drfrostmaths.com/keyskills.php?permid=118
Multiples of Surds

and

6 √20=𝟔 √ 𝟒 √ 𝟓=𝟏𝟐
? √𝟑
7 √ 12=𝟕 √𝟒 √𝟑=𝟏𝟒
? √𝟑
2 √ 45=𝟐 √ 𝟗 √?𝟓=𝟔 √𝟓
Test Your Understanding

a
2 √ 75=𝟐 √𝟐𝟓 √𝟑=𝟏𝟎
? √𝟑
b 3 √ 40=𝟑 √𝟒 √𝟏𝟎=𝟔
? √𝟏𝟎
c 4 √ 48=𝟒 √𝟏𝟔 √𝟑=𝟏𝟔
? √𝟑
d 3 √ 200=𝟑 √ 𝟏𝟎𝟎 √?𝟐=𝟑𝟎 √ 𝟐
e
5 √ 45=𝟓 √𝟗 √ 𝟓=𝟏𝟓
? √𝟓

Practise this specific Key Skill:

https://www.drfrostmaths.com/keyskills.php?permid=796
Multiplying Surds

√3× √5=√𝟏𝟓
?
? We can multiply surds together,
e.g. , but we can’t combine a surd
? and a non-surd together – they
must remain separate (with non-
? surd first)

?
Multiplying non-surds:

2 √ 3 ×2 √ 5=𝟒
? √ 𝟏𝟓 Multiplying surds:

3 √ 2 ×3 √ 2=𝟏𝟖
? and

√ 18 × 4 √ 2=𝟐𝟒
? and
Test Your Understanding

6× √ 7=𝟔 √𝟕
a ?
b ?
c ?
d ?
e ?
f ?
g ?
h ?
Practise this specific Key Skill:
https://www.drfrostmaths.com/keyskills.php?permid=121
https://www.drfrostmaths.com/keyskills.php?permid=122
Exercise 1
1 Simplify the following: 4 Simplify the following:
a ? a ?
? following:
b Simplify the b ? as a
c ? Express the following
single square root?
c
d
Simplify
? following:
the ?
d (hint: do the steps of simplification
e ? backwards!)
5
2
a ? Express the following as a
b ? single square root:
c ? a ?
d ? b ?
e ? c ?
3 d ?
a ?
b ? 6
c ? a ?
d ? b ?
e ?
f ?
g ?
Adding Surds

√ 3+ √ 3=𝟐 √? 𝟑
Think of it as “if I have one lot of and I add another lot of , I
have two lots of ”.
It’s just how we collect like terms in algebra, e.g.

2 √ 5+√ 5=𝟑 √ 𝟓
?
?
?

?
Test Your Understanding

√√√ √
?

𝟑
a

3+ 3+ 3=𝟑
b ?
c
?
d
?

Practise this specific Key Skill:

https://www.drfrostmaths.com/keyskills.php?permid=119
https://www.drfrostmaths.com/keyskills.php?permid=120
Brackets and Surds

√ 2 ( 3+ √ 2 )= 𝟑 √?𝟐+ 𝟐
( √ 2+ 1 ) ( √ 2− 1 )=𝟐 + √ 𝟐 − √
? 𝟐 −𝟏=𝟏

( √8+3 )( √2+5)=√𝟏𝟔+𝟓 √𝟖+𝟑 √𝟐+𝟏𝟓

?

2
( √5−2) = (√ 𝟓−𝟐)(√ 𝟓−𝟐)
?
Test Your Understanding

5√ (2+ √ 3)=𝟐 √𝟓+√ 𝟏𝟓

a ?
b ?
c ?
d ?

N
𝐴𝑟𝑒𝑎=𝟔+𝟓? √ 𝟑
3+ √ 3

Practise these specific Key Skills:

https://www.drfrostmaths.com/keyskills.php?permid=125
https://www.drfrostmaths.com/keyskills.php?permid=311
https://www.drfrostmaths.com/keyskills.php?permid=313
1+3 √ 3 https://www.drfrostmaths.com/keyskills.php?permid=314
Exercise 2
1 Simplify the following: 3 Expand and simplify:
a ? a ?
b ? b
Determine the area of :
?
c ?
Expand and simplify the following, c ?
leaving your answers in?the form ?
d d
e ? e ?
f ?
f ?
2 4
c
a √3 b

4+3 √5
√5−1
√5

6−√5
?
2+√3
a
b ? √ 5+ 3
c ?
d ? 𝑨=𝟐 √ 𝟑+𝟑
? 𝑨=𝟏+?√ 𝟓 𝑨=𝟓+𝟓
? √𝟓
e ? 5 𝑃
Find the length of .
√7− 2 (Using Pythagoras)
?
√ 7+𝑄
2
Rationalising The Denominator
Here’s a surd. What could we multiply it by such that it’s no
longer an irrational number?

√ 5 × √?5=5?
1 √ 2 √2
× ? = ?
√2 √2 2
In this fraction, the denominator is Fro Side Note: There’s two reasons
irrational. ‘Rationalising the why we might want to do this:
denominator’ means making the 1. For aesthetic reasons, it makes
denominator a rational number. more sense to say “half of root 2”
rather than “one root two-th of
What could we multiply this fraction by to 1”. It’s nice to divide by
both rationalise the denominator, but something whole!
leave the value of the fraction 2. It makes it easier for us to add
unchanged? expressions involving surds.
More Examples
Test Your Understanding:

?
a ?

?
b ?

?
c ?
?

?
Practise this specific Key Skill:
https://www.drfrostmaths.com/keyskills.php?permid=126
More Complex Denominators
You’ve seen ‘rationalising a denominator’, the idea being that we don’t like to
divide things by an irrational number.

But what do we multiply the numerator and denominator by if we have a more

complicated denominator?

? ?

We multiply the denominator by what is known as its

conjugate, i.e. the same expression but the + replaced with
and vice versa. That way, we obtain the difference of two
squares. Since , any surds will be squared and thus we’ll end
up with no surds in the denominator.
And if we’ve multiplied the denominator by this, we need to
multiply the numerator by the same to preserve the value.
More Examples

3
× √ 6 +2 = 3 √ 6 +6 You can explicitly expand out in
the denominator, but remember
? ?
√6 −2 √ 6 +2 2 that so we get
Just remember: ‘difference of
two squares’!

4 √ 3 − 1 4 √3 − 4
=2 √?3 −2
× ? = ?2
√ 3+1 √3 − 1

3 √ 2+4 𝟓 √ 𝟐+𝟕
? 𝟑𝟎+𝟐𝟏 √ 𝟐+𝟐𝟎 √ 𝟐+𝟐𝟖
× = ?
5 √ 2− 7 𝟓 √ 𝟐+𝟕 𝟏
Test Your Understanding
Rationalise the
denominator and simplify
? √𝟓
𝟖+𝟒

Rationalise the denominator and

simplify
AQA FM June 2013 Paper 1
Solve
Give your answer in the form where

𝟐 √𝟑−𝟏 ? 𝟑 √𝟑−𝟏
and are integers.

× 𝟖 ? √ 𝟑+𝟏
𝟑 √ 𝟑+𝟏 𝟑 √𝟑−𝟏 𝒚= ×
√𝟑−𝟏 √ 𝟑+𝟏
Exercise 3
1 Rationalise the denominator 2 Expand and simplify:
and simplify the following: ?
a ? 3
Rationalise the denominator, giving
your answer in the form .
b ? ?
Solve giving your answer in the form .
c ? 4
Solve
d ? ?
Simplify:
5
e ?
?
6

?
A final super hard puzzle

N
Solve

But